Clusters in Digital Pictures

As a generalization of the notion of connectedness, we define a set S to be a D-cluster if one can move from any point of S to any other along a path of points in S on which successive points are at most distance D apart. We define (in two ways) the D-hull of a set S , and show that S is a D -cluster if and only if its D-hull is connected. This allows us to define holes in a cluster ( S has “ D -holes” if its D -hull has holes), and we then show that for the two definitions of D -hull, these holes are in one-to-one correspondence. We also discuss possible ways of defining the D-border of a D -cluster.